Abstract—Parkinson's disease (PD) is a heterogeneous movement disorder that often appears in the elderly. PD is induced by a loss of dopamine secretion. Some drugs increase the secretion of dopamine. In this paper, we will simply study the stability of PD models as a nonlinear delay differential equation. After a period of taking drugs, these act as positive feedback and increase the tremors of patients, and then, the differential equation has positive coefficients and the system is unstable under these conditions. We will present a set of suggested modifications to make the system more compatible with the biodynamic system. When giving a set of numerical examples, this research paper is concerned with the mathematical analysis, and no clinical data have been used. Keywords—Parkinson's disease, stability, simulation, two delay differential equation. I. INTRODUCTION D is a sort of disorder of movement. It occurs when the brain's nerve cells do not contain enough of a brain chemical called dopamine. Gait and balance disorders like PD have a common effect on the geriatric population. It is inherited at times, but most cases do not seem to occur within families. Environmental exposure to chemicals may play a part. Dopamine plays a key function in controlling body movement. A decrease in dopamine is responsible for many of the PD symptoms. It is unknown exactly what causes the loss of nerve cells. Most experts agree that blame rests on a combination of genetic and environmental factors [1]. PD was modeled mathematically in 1961 [2]), after it was observed that the tremors were affected by the emotional state of the patient. In this model, the Van der Pol model was used. This model relied on two parts, one of which acts as a positive feedback and the other as negative feedback, or the two terms act as negative feedback for the tremor [2]. In 2009, Lainscsek et al. used delay differential equations to describe PD and used time series analysis to estimate the constants in the differential equation, which were two delays, one for the control - on and the other for the control – off state [3]. Claudia [3] noticed repeated tapping movements of the finger and assumed a periodic function as: ) ft cos( ) t ( x  Mohamed Elfouly is with the Mansoura University, Egypt (e-mail: maf080877@yahoo.com). Then, the velocity: ) ft sin( f ) t ( x                       f t f cos f ) t ( x 2          t f cos a ) t ( x       t ax t x ) (  Hence, the model was deduced from this symptom of PD. In the end, any symptoms of the disease are caused by a defect in the transmission of the nerve signal to the locomotor system. From the results of [2] and [3], it can be assumed that the defect in Parkinson's patients can be expressed by the delay differential equation with positive and negative feedback [4] as: 0 < t < - < - and 0 > t , h = x(t) ) - )x(t - x(t a + ) - x(t a + ) - x(t a = dt dx(t) 1 2 2 1 3 2 2 1 1       (1) where 3 2 1 , , a a a are constants, ) (t x the positive function that describes the defects in the locomotor system in Parkinson's patients, in [2] ) (t x express the amplitude of the tremor, h the initial value for function ) (t x , 1  the delay of control-on and 2  the delay of control-off, 2 , 1 , 2   i f i i   , i f the frequency of the Oscillatory movement. Delays depend on the time the drugs take effect on and off and the emotional state of the patient. The nonlinear part adjusts for any perturbation in the model. In the second section, we will do a simple analysis of the stability of this assumed model and determine the equilibrium points. In 2019, Ahmed [5] assumed that all the constants are 3 2 1 , , a a a positive with a positive initial value h , depending on a set of positive values that appeared when Claudia's estimation of the constants  ଵ , ଶ , ଷ [3]. Ahmed presented modification to make the system stable at a non-zero equilibrium point. In the third and fourth sections, we will present other modifications with a simple stability analysis for these modifications. Stability Analysis of Two-delay Differential Equation for Parkinson's Disease Models with Positive Feedback M. A. Sohaly, M. A. Elfouly P World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:15, No:1, 2021 8 International Scholarly and Scientific Research & Innovation 15(1) 2021 ISNI:0000000091950263 Open Science Index, Mathematical and Computational Sciences Vol:15, No:1, 2021 publications.waset.org/10011816/pdf